Extended Gini-Type Measures of Risk and Variability

Abstract

The aim of this paper is to introduce a risk measure that extends the Gini-type measures of risk and variability, the Extended Gini Shortfall, by taking risk aversion into consideration. Our risk measure is coherent and catches variability, an important concept for risk management. The analysis is made under the Choquet integral representations framework. We expose results for analytic computation under well-known distribution functions. Furthermore, we provide a practical application. JEL classification: C6, G10a prominent trend associated with tail-based risk measures has emerged, especially with the most popular ones nowadays: the Value-at-Risk (VaR) and the Expected Shortfall (ES). However, this kind of risk measures does not capture the variability of a financial position, a primitive but relevant concept. In order to solution this issue, some authors have proposed and studied specific examples of risk measures.In this sense, Fischer [8] considered combining the mean and semi-deviations. Regarding tail risk, Furman and Landsman [10] proposed a measure that weighs the mean and standard deviation in the truncated tail by VaR, while Righi and Ceretta [17] considered penalizing the ES by the dispersion of losses exceeding it. From a practical perspective, Righi and Borenstein [18] explored this concept, calling the approach as loss-deviation, for portfolio optimization. In a more general fashion, Righi [19] presents results and examples about compositions of risk and variability measures in order to ensure solid theoretical properties.Recently, Furman et al. [11] introduced the Gini Shortfall (GS) risk measure which is coherent and satisfies co-monotonic additivity. GS is a composition between ES and tail based Gini coefficient. However, GS supposes that all individuals have the same attitude towards risk, while agents differ in the way they take personal decisions that involve risk because of discrepancies in their risk aversion. To incorporate such psychological behavior in tail risk analysis, we introduce a generalized version of the GS. This risk measure, called Extended Gini Shortfall (EGS), captures the notion of variability, satisfies the co-monotonic additivity property, and it is coherent under a necessary and sufficient condition for its loading parameter. The consideration of the decision-maker risk aversion, joined to these properties, is in consonance to what agents seek when searching for a suitable measure of risk. The approach followed in this article leads us to a new family of spectral risk measures, proposed by Acerbi [1], with an attractive weighting function.In this sense, we discuss, in a separated manner, the properties from the variability term and our composed risk measure. Moreover, we discuss in details the role of each parameter in the mentioned weighting function. Furthermore, we expose results on analytic formulations for computation of EGS under known distribution functions. Our focus in this paper is on theoretical results, but this approach gives rise to further forthcoming inve...

A composition between risk and deviation measures

“…There are other examples in the literature of functionals composed by a coherent risk measure and a non-convex deviation that is again a coherent risk measure. This is exactly what happens for the Tail Gini Shortfall, proposed by Furman et al (2017), and its extension introduced in Berkhouch et al (2017). The idea of such risk measures is to have a composition of the form ρ + βD between ES and a Gini functional restricted to the distribution tail.…”

“…Righi and Ceretta (2016) considered penalizing the ES by the dispersion of results that represent losses exceeding the ES. Furman et al (2017) and Berkhouch et al (2017) penalize ES by the dispersion of tailbased Gini measures. These risk measures are individual examples, rather than a general approach.…”

Spectral Risk Measures and Uncertainty

“…• Mean Gini coefficient (Gini): This is a statistical coefficient that measures the variation degree in the set of values X(ω) as ω varies in Ω, see [Shalit and Yitzhaki, 1984], [Giorgi, 1993], [Giorgi, 2005], [Yitzhaki, 1998], [Ceriani and Verme, 2012], [Furman et al, 2017] and [Berkhouch et al, 2018]. It is defined as:…”

“…In this section we are going to provide an illustration using a case study. We consider a scenario-based SRM, ρ φ , where the spectral risk measure ρ s φ is the Extended Gini Shortfall introduced in [Berkhouch et al, 2018] conform:…”

Worst-case analysis of Gini mean difference safety measure

“…Furman et al [13] proposed the Gini shortfall risk measure and explored its advantages over the other existing risk measures. Recently, Berkhouch et al [4] introduced extended Gini shortfall, which encompassed in it the properties of Gini shortfall as well as variability in the risk-taking capacity of investors.…”